## Tesi etd-07142011-235540 |

Thesis type

Tesi di dottorato di ricerca

Author

AL-HASSEM, NAYAM

URN

etd-07142011-235540

Title

Shape optimization problems of higher codimension

Settore scientifico disciplinare

MAT/05

Corso di studi

MATEMATICA

Supervisors

**tutor**Buttazzo, Giuseppe

Parole chiave

- p-Laplacian equation
- Location problems.
- Network problems
- Gamma-convergence
- Compliance functional
- Asymptotic shapes
- Stazionary configuration

Data inizio appello

18/07/2011;

Consultabilità

Completa

Riassunto analitico

The field of shape optimization problems has received a lot of attention in recent

years, particularly in relation to a number of applications in physics and engineering

that require a focus on shapes instead of parameters or functions. In general for ap-

plications the aim is to deform and modify the admissible shapes in order to optimize

a given cost function. The fascinating feature is that the variables are shapes, i.e.,

domains of R^{d}, instead of functions. This choice often produces additional dicul-

ties for the existence of a classical solution (that is an optimizing domain) and the

introduction of suitable relaxed formulation of the problem is needed in order to get a

solution which is in this case a measure. However, we may obtain a classical solution by

imposing some geometrical constraint on the class of competing domains or requiring

the cost functional verifies some particular conditions. The shape optimization problem

is in general an optimization problem of the form

min\{F(\Omega): \Omega \in {\cal O} \};

where F is a given cost functional and {\cal O} a class of domains in R^{d}. They are many

books written on shape optimization problems. The thesis is organized as follows: the

first chapter is dedicated to a brief introduction and presentation of some examples.

In Academic examples, we present the isoperimetric problems, minimal and capillary

surface problems and the spectral optimization problems while in applied examples the

Newton's problem of optimal aerodinamical profile and optimal mixture of two con-

ductors are considered. The second chapter is concerned with some basics elements of

geometric measure theory that will be used in the sequel. After recalling some notions

of abstract measure theory, we deal with the Hausdorff measures which are important

for defining the notion of approximate tangent space. Finally we introduce the notion

of approximate tangent space to a measure and to a set and also some differential op-

erators like tangential differential, tangential gradient and tangential divergence. The

third chapter is devoted to the topologies on the set of domains in R^{d}. Three topologies

induced by convergence of domains are presented namely the convergence of charac-

teristic functions, the convergence in the sense of Hausdorff and the convergence in

the sense of compacts as well as the relationship between those different topologies. In

the fourth chapter we present a shape optimization problem governed by linear state

equations. After dealing with the continuity of the solution of the Laplacian problem

with respect to the domain variation (including counter-examples to the continuity and

the introduction to a new topology: the

gamma-convergence), we analyse the existence of

optimal shapes and the necessary condition of optimality in the case where an optimal

shape exists. The shape optimization problems governed by nonlinear state equations are treated in chapter ve. The plan of study is the same as in chapter four that

is continuity with respect to the domain variation of the solution of the p-Laplacian

problem (and more general operator in divergence form), the existence of optimal

shapes and the necessary condition of optimality in the case where an optimal shape

exists. The last chapter deals with asymptotical shapes. After recalling the notion

of Gamma-convergence, we study the asymptotic of the compliance functional in different

situations. First we study the asymptotic of an optimal p-compliance-networks which

is the compliance associated to p-Laplacian problem with control variables running in

the class of one dimensional closed connected sets with assigned length. We provide

also the connection with other asymptotic problems like the average distance problem.

The asymptotic of the p-compliance-location which deal with the compliance associ-

atied to the p-Laplacian problem with control variables running in the class of sets of

finite numbers of points, is deduced from the study of the asymptotic of p-compliance-

networks. Secondly we study the asymptotic of an optimal compliance-location. In

this case we deal with the compliance associated to the classical Laplacian problem

and the class of control variables is the class of identics n balls with radius depending

on n and with fixed capacity.

years, particularly in relation to a number of applications in physics and engineering

that require a focus on shapes instead of parameters or functions. In general for ap-

plications the aim is to deform and modify the admissible shapes in order to optimize

a given cost function. The fascinating feature is that the variables are shapes, i.e.,

domains of R^{d}, instead of functions. This choice often produces additional dicul-

ties for the existence of a classical solution (that is an optimizing domain) and the

introduction of suitable relaxed formulation of the problem is needed in order to get a

solution which is in this case a measure. However, we may obtain a classical solution by

imposing some geometrical constraint on the class of competing domains or requiring

the cost functional verifies some particular conditions. The shape optimization problem

is in general an optimization problem of the form

min\{F(\Omega): \Omega \in {\cal O} \};

where F is a given cost functional and {\cal O} a class of domains in R^{d}. They are many

books written on shape optimization problems. The thesis is organized as follows: the

first chapter is dedicated to a brief introduction and presentation of some examples.

In Academic examples, we present the isoperimetric problems, minimal and capillary

surface problems and the spectral optimization problems while in applied examples the

Newton's problem of optimal aerodinamical profile and optimal mixture of two con-

ductors are considered. The second chapter is concerned with some basics elements of

geometric measure theory that will be used in the sequel. After recalling some notions

of abstract measure theory, we deal with the Hausdorff measures which are important

for defining the notion of approximate tangent space. Finally we introduce the notion

of approximate tangent space to a measure and to a set and also some differential op-

erators like tangential differential, tangential gradient and tangential divergence. The

third chapter is devoted to the topologies on the set of domains in R^{d}. Three topologies

induced by convergence of domains are presented namely the convergence of charac-

teristic functions, the convergence in the sense of Hausdorff and the convergence in

the sense of compacts as well as the relationship between those different topologies. In

the fourth chapter we present a shape optimization problem governed by linear state

equations. After dealing with the continuity of the solution of the Laplacian problem

with respect to the domain variation (including counter-examples to the continuity and

the introduction to a new topology: the

gamma-convergence), we analyse the existence of

optimal shapes and the necessary condition of optimality in the case where an optimal

shape exists. The shape optimization problems governed by nonlinear state equations are treated in chapter ve. The plan of study is the same as in chapter four that

is continuity with respect to the domain variation of the solution of the p-Laplacian

problem (and more general operator in divergence form), the existence of optimal

shapes and the necessary condition of optimality in the case where an optimal shape

exists. The last chapter deals with asymptotical shapes. After recalling the notion

of Gamma-convergence, we study the asymptotic of the compliance functional in different

situations. First we study the asymptotic of an optimal p-compliance-networks which

is the compliance associated to p-Laplacian problem with control variables running in

the class of one dimensional closed connected sets with assigned length. We provide

also the connection with other asymptotic problems like the average distance problem.

The asymptotic of the p-compliance-location which deal with the compliance associ-

atied to the p-Laplacian problem with control variables running in the class of sets of

finite numbers of points, is deduced from the study of the asymptotic of p-compliance-

networks. Secondly we study the asymptotic of an optimal compliance-location. In

this case we deal with the compliance associated to the classical Laplacian problem

and the class of control variables is the class of identics n balls with radius depending

on n and with fixed capacity.

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